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# What Is the Pizza Problem in Psychology Research?

## The pizza problem helps explain a core concept in approaches to research.

The pizza problem comes from a tweet about how to get the most food for an order of pizza.
Source: Photo by Vincent Rivaud on Pexels.

Olivia Guest and Andrea Martin open a brilliant new paper with a tweet about pizza. The tweet asks a simple question: imagine you can get two 12-inch pizzas or one 18-inch pizza. Which one gives you the most food?

Guest and Martin apply mathematical modeling to psychology, meaning that they write formal mathematical equations that can be used to explain observed psychological data. Most of psychology tends to talk about whether there is or is not an effect—by which they mean that one variable (like being more open to new experiences) is related to another (like being more willing to listen to people you disagree with), or whether a treatment for depression did or didn’t improve symptoms. The mathematical modeling approach would try to come up with equations that explain whatever effect was seen more precisely. (If you read last week’s post, you’ll know that I’ve been inspired to think and read about mathematical modeling after seeing how sophisticated—and useful—modeling has been in guiding out pandemic response.)

Back to pizza. The intuitive answer to the question above is the two pizzas. If you aren’t using a mathematical model, you would just eyeball it and say “well, two is more than one, and 18 isn’t even close to two times 12.” But we can write a simple mathematical model to answer this question.

Assume that a pizza can be modeled as a circle. (Pizzas aren’t perfectly circle shaped, but they’re close enough. Assumptions like this that make the math easier are an important part of being able to get going with models.) We know the area of a circle is pi multiplied by r2, where r is the radius—half of the full length of the pizza. Using that formula, we find that a 12-inch pizza gives 113 square inches of food. Add up two of those, and you find that your first order gives 226 square inches of food. Meanwhile, the one 18-inch pizza gives 254 square inches of food. Getting the one big pizza actually gives you more food!

These stock photo people are about to be surprised at how little pizza they got!
Source: Photo by Fauxels on Pexels.

This example isn’t meant to help you plan your ‘za procurement strategy. It’s meant to illustrate how writing down formal equations, even when they are simple abstractions, can give you surprising results. Equations make precise what a theory thinks is happening—and should happen in the future. Leaving theory at the level of words opens scientists up to the pizza problem. One answer might sound intuitive and plausible, but if you took the time to work out the math behind the intuition, you’d find that it was wrong. Scientists, like me and most other psychologists, who don’t typically use formal equations are leaving themselves open to the pizza problem.

Guest and Martin’s article pairs well with another recent article on modeling by Iris van Rooij and Giosue Baggio. In it, they argue that an important first step in research on a particular topic is just doing the modeling. As we saw in the pizza problem, the math can yield surprising results. Sometimes, it yields results that are totally wild—or, as they put it, implausible. By working through several different possible mathematical descriptions of a subject up front, van Rooij and Baggio argue that psychologists can figure out which explanations seem plausible, and focus the experiments and studies they design on testing these. The explanations that math shows lead to totally wild, implausible results can be ignored (or at least put on a back burner).

The pizza problem leads to the pizza insight!
Source: Photo by ThisIsEngineering on Pexels.

Both of these approaches take advantage of an insight that people who work in math often have, what we might call the “pizza insight.” The pizza insight is that math isn’t just dry, rote calculation; math is also a source of excitement and surprise. Math can defy our intuitions and lead to unexpected insight and profundity. When you accept the results of the pizza problem as suggesting the need for using mathematical tools, you open yourself up to the pizza insight, and to the ability to be dazzled and delighted by math.

As a researcher who has already completed his Ph.D. and is in his 30s, I find it frustrating (and even a bit threatening) to think that I don’t have the tools needed to do the kind of research I think psychology most needs. I can understand a push back against emphasizing mathematical modeling, and I don’t mean to say that people without these skills can’t contribute to our science. But I think that many (dare I say most) researchers can at least learn to understand and interact with these models, and to use their insights when designing their next study. Not all public health officials need to be able to develop a model of disease spread. But the good ones will listen carefully to the briefings created by the modelers, and will take next steps with these model results in mind. Having the pizza insight means allowing your intuitions to be humbled by the math, even if it’s not math you worked out, and appreciating its unexpected insights.

References

Guest, O., & Martin, A. E. (2020). How computational modeling can force theory building in psychological science. Preprint on PsyArXiv. https://psyarxiv.com/rybh9/

van Rooij, I., & Baggio, G. (2020). Theory before the test: How to build high-verisimilitude explanatory theories in psychological science. Preprint on PsyArXiv. https://psyarxiv.com/7qbpr/

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